AI 中文总结
该研究推导有限同形谱量子奥托引擎的功分布等闭式表达式，区分功不可靠性的多类贡献，表明最大平均输出与最大无量纲可靠性选不同工作点。
AI 中文摘要
同形量子奥托引擎中，所有布居的能隙均按同一因子缩放，该模型是参考模型，其准静态随机效率与轨迹无关，而功仍具有涨落性。针对任意有限同形谱，我们推导了两点测量的功分布，并将前两个功矩约化为端点能量矩。针对均匀间距的能级梯，我们得到了完整功分布、平均功、方差及信号-宽度可靠性的有限N闭式表达式。该能级梯连接了量子比特与谐振子极限，揭示了有限N可靠性交叉现象，并表明高温与无限维极限不可对易。这种不可对易性反映了有界与无界谱的区别：在固定有限N下，吉布斯态具有可归一化的无限高温极限，而谐振子则保留不断扩展的热尾。我们用精确公式比较了标准平均输出方案与功可靠性，表明最大平均输出与最大无量纲可靠性选择不同的工作点。该基准扩展至不完全对角重置及由跃迁矩阵描述的有限时间幺正冲程，以有限能级梯协议和谐振子突然切换振荡器基准作为受控示例。我们对精确同形性的弱偏差进行了微扰处理，表明与能级相关的能隙畸变如何重新引入准静态效率涨落并改变功可靠性。这些结果共同区分了量子奥托引擎中功不可靠性的有限尺寸、不完全热化、有限时间及弱谱畸变贡献。
英文摘要
Homothetic quantum Otto engines---where all populated energy gaps are rescaled by a common factor---provide a reference model in which the quasistatic stochastic efficiency is trajectory-independent while work remains fluctuating. For arbitrary finite homothetic spectra we derive the two-point-measurement work distribution and reduce the first two work moments to endpoint energy moments. Specializing to a uniformly spaced ladder gives closed finite-$N$ expressions for the full work distribution, mean work, variance, and signal-to-width reliability. This ladder connects the qubit and oscillator limits, reveals a finite-$N$ reliability crossover, and demonstrates that the high-temperature and infinite-dimensional limits do not commute. The noncommutation reflects a bounded-versus-unbounded spectral distinction: at fixed finite $N$ the Gibbs state has a normalizable infinite-temperature limit, whereas the oscillator retains an ever-expanding thermal tail. The exact formulas are used to compare standard mean-output prescriptions with work reliability, showing that maximum mean output and maximum dimensionless reliability select different operating points. The benchmark is extended to incomplete diagonal reset and to finite-time unitary strokes described by transition matrices, with a finite-ladder protocol and a harmonic sudden-switch oscillator benchmark as controlled examples. Weak deviations from exact homothety are treated perturbatively, showing how level-dependent gap distortions reintroduce quasistatic efficiency fluctuations and modify work reliability. Together, these results separate finite-size, incomplete thermalization, finite-time, and weak spectral-distortion contributions to work unreliability in quantum Otto engines.
